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A cute way of looking at angles

May 06, 2012

A cute way of looking at angles

“Life is really simple, but we insist on making it complicated.” - Confucius

Learning is a process of discovery: some discoveries are given; some are made. Trigonometry is one where the ideas can be easy to grasp or an impossible abstraction.

Let us look at the wrong way to understand this. Technically, sin x can be defined as x – (x*x*x)/(3*2*1) (x*x*x*x*x)/(5*4*3*2*1) (x*x*x*x*x*x*x)/(7*6*5*4*3*2*1) ... well, it doesn’t really matter. The fractions will get successively smaller and they go on literally forever as it is an infinite series. To children and for many adults, this is as clear as mud.

My husband was telling me how he worked out the basics of sine and cosine when he was about 9. When playing in the garden his father had planted peas and there were some pea sticks left over – a simple length of wood. He was holding it in the middle and twirling it, just as children will play with anything. He noticed that when it was held straight up and down, it was as tall as it could be. When held flat, it was as long as it could be but not at all tall. When it was at about 45 degrees, it was about half as tall and half as long as it could be – the stick was shared between tall and long. A quarter of a turn didn’t give quite the split between tall and long as he expected. There was string for tying up the peas handy so it was easy to measure. After some puzzling, he worked out that the fraction of height versus length changed for every angle and that a really smart person could write down all the numbers. When recalling the story my husband was still obviously very disappointed having learned that the ancient Greeks had beaten him to the idea by quite a few years.

Allowing or encouraging children to 'discover' how the world works is the best form of education there is. Their emotional investment of the investigation far outweighs any classroom based learning.

A child who discovers something by them self will always remember it much as my husband did.